# Is there a good concept of a measurable fibration?

In probability theory, there are many results which are valid in purely measurable settings, usually beginning with the assumption, "let $(\Omega, \mathcal F, \mathbb P)$ be an abstract probability space." On the other hand, many results require a more "locally measurable" structure, often taking the form of a Borel $\sigma$-algebra on a topological space. In an effort to compromise between the two concepts, I am wondering if there is a good notion of a measurable fibration.

Let $E$ be a measurable space, let $B$ be a topological space, and let $\pi : E \to B$ be a Borel-measurable function. I would like to define a "measurable homotopy lifting property" with respect to an abstract probability space $(\Omega, \mathcal F, \mathbb P)$.

Let $I = [0,1]$ be the unit interval, equipped with its Borel $\sigma$-algebra $\mathcal B(I)$ and Lebesgue measure $\lambda$. A "measurable homotopy" should be a measurable map $f : \Omega \times I \to B$, where the product $\Omega \times I$ is equipped with the tensor product $\sigma$-algebra $\mathcal F \otimes \mathcal B(I)$. Measures push forward, so this naturally endows $B$ with a probability measure $P_f = f_*(\mathbb P \otimes \lambda)$.

Normally, one next considers a lift $\tilde f_0 : \Omega \to E$ of the map $f_0 = f|_{\Omega \times \{0\}}$. This makes sense in topology, because by lifting at a point one can "tug" the rest of the homotopy up to $E$. However, this doesn't make sense in this context: the set $\Omega \times \{0\}$ has measure zero, hence is meaningless from the point of view of measure theory. Hence:

Question: is there a generalization of the homotopy lifting property to this measurable setting?

Even thought $0 \in I$ has no special meaning probabilistically, the concept of a random number $\iota \in I$ with distribution $\lambda$ does make sense. In fact, the product measure $\mathbb P \otimes \lambda$ represents choosing a random $\omega \in \Omega$ and random $\iota \in I$ independently from one another. Consequently, a random point $(\omega, \iota)$ picks out a particular function $f_{\iota} := f|_{\Omega \times \{\iota\}}$. Diagrammatically this results in a mess of arrows, so I'll stop the speculation and leave the question to the community.

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You could consider $[0,\epsilon] \hookrightarrow [0,1]$ instead of $\lbrace0\rbrace \hookrightarrow [0,1]$. Homotopically it's the same, and possibly more what you want: the map from the small interval seems to be a better choice of 'initial data' for the lift. –  David Roberts Jan 30 '13 at 23:59
Or perhaps it's better to think about what you want trivial cofibrations to be first, and then define fibrations to be those with the right lifting property against those... –  David Roberts Jan 31 '13 at 0:00
I'm ill-equipped to understand the topological part of the picture, but in Dan Rudolph's book, Fundamentals of Measurable Dynamics, he spends quite a lot of time doing a fibre construction in Lebesgue spaces. –  Anthony Quas Jan 31 '13 at 0:09